Crow was 1-4 with a 4.06 ERA as a freshman at the University of Missouri and allowed 94 hits in 77 2/3 IP for a .301 batting average. In his sophomore season, Aaron was 9-4 with a 3.59 ERA, finishing third on his team in ERA. Crow emerged as a top prospect in the 2007 Cape Cod League, going 3-1 with a 0.69 ERA for the Falmouth Commodores. He allowed only 19 hits and 9 walks in 40 1/3 IP while striking out 36. It tied Tim Lincecum for the 10th-lowest ERA in league history. He was named to the All-Star team and Baseball America rated him the top prospect in the circuit.

The Nationals offered $3.5 million, while Crow held out for $4 million. When he was unable to come to terms with the Nationals, Crow signed with the independent Fort Worth Cats to wait until being drafted again in 2009. For the Cats in 2008, Crow pitched one inning, giving up one hit and no runs. He was 3-0 with a 1.06 ERA for Fort Worth in 2009. Crow was then picked by the Kansas City Royals 12th overall in the 2009 amateur draft, a drop of 3 spots from 2008; some projections had him going as high as #4 before his stock fell. He was signed by scout Scott Melvin to a $3 million dollar big league contract that included a $1.5 million bonus.

Aaron Crow made his major league debut on March 31, 2011, in relief of Luke Hochevar on Opening Day facing the Los Angeles Angels of Anaheim. He had made the team with an outstanding spring, during which he gave up only 3 runs in 13.1 innings. In his debut, he came in with a man on base and two outs in the 6th inning, and struck out the first batter he faced, Howie Kendrick, to get out of the inning; he then followed with a perfect 7th inning, striking out two more batters, in a very impressive outing. He was followed on the mound by two other relievers making their debut that day, Nathan Adcock and Tim Collins and both pitched scoreless innings, although the Royals lost, 4-2. Crow went on to be the Royals' sole representative at the 2011 All-Star Game. He finished the year with a record of 4-4, 2.76, in 57 games, having given up 55 hits and 31 walks in 62 innings, while striking out 65 batters. He was exclusively used as a set-up man for closer Joakim Soria.

In 2012, the Royals headed into spring training with plans to make Crow a starter. However, when Soria was diagnosed with an elbow injury in mid-March, they turned around and instead announced that Aaron would remain in the bullpen, where he was to compete with Jonathan Broxton and Greg Holland for the closer's job in Soria's expected absence. Holland won the competition and in turn became one of the best closers in the majors, but Aaron had a solid year, making 73 appearances with a record of 3-1, 3.48 and a couple of saves. In 2013, he went 7-5, 3.38 in 57 games, but pitched only 48 innings, as the Royals' bullpen became crowded with the development of Holland, the arrival of Wade Davis and solid contributions by others such as Tim Collins, Kelvin Herrera and Luke Hochevar. That bullpen was one of the reasons the Royals had one of their best seasons in year, and why they finally made it over the hump in 2014, going all the way to the World Series. It was bittersweet for Aaron however, as his role became even more marginal. He did make 67 appearances, with a record of 6-1 and 3 saves, but his ERA rose to 4.12, and he did not pitch at all in the postseason, being displaced in his role by rookie Brandon Finnegan, pitching in his first professional season.